Illustrated Physics
physics, drawn
Concept

Binding energy — where it appears

The energy that must be supplied to take a bound system apart, and therefore the amount by which the whole weighs less than its parts. Helium-4 is 28.3 MeV lighter than two protons and two neutrons, which is seven parts in a thousand of its own mass.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones.

The mass that is missing

A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

relativity · Mass-energy The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales.

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

relativity · Mass-energy Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more.

Why an atom is the size it is

A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

quantum · Atomic structure An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium.

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

waves · Attenuation The month a binding energy would bend. The Moon's orbit seen with the Sun held off to the right, drawn as it would be if the Earth's binding energy fell towards the Sun more weakly than the rest of it. The Earth is bound by 4.5 × 10⁻¹⁰ of its mass-energy and the Moon by 1.9 × 10⁻¹¹, so with the Sun pulling at 5.93 mm/s² the Moon is pushed sunward relative to the Earth by 2.6 × 10⁻¹² m/s² for every unit of η. That push turns once a synodic month relative to the orbit, 29.53 days, and Hill's equations about a circular orbit — integrated from the forced solution and held on it to 4 × 10⁻¹¹ over twelve months — give a radial displacement of 8.0 m times η times the cosine of the lunar phase: outward at new moon, inward at full. The complete lunar theory, with the Sun's tide on the orbit included, gives 13.1 m. The displacement is drawn about 7 × 10⁶ times larger than it would be at η = 1.

The binding energy that has to fall too

Every laboratory test of the equivalence principle compares bodies whose own gravity is a part in 10²⁵ of their mass, so none of them can ask whether gravitational binding energy falls like everything else. The Earth is bound by five parts in ten billion and the Moon by twenty times less, and if that difference fell differently the Moon's orbit would lean towards the Sun once a month — by a distance lasers have been measuring since 1969.

astrophysics · Equivalence principle The ripple that sits on top of an absorption edge. The Cu K absorption of copper foil, 293 K through its edge and for seven hundred electronvolts above it, with the smooth atomic background it would have if the absorbing atom were alone drawn beneath it. The difference between the two is the fine structure: a modulation reaching 11 per cent, dying away as the photon energy rises, and entirely absent from a free atom. It is there because the ejected electron is a wave that the neighbouring atoms scatter back onto the atom that emitted it, so the absorption depends on whether the returning wave arrives in step with the outgoing one — which depends on the distance to the neighbour and on nothing else about the sample.

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

waves · Attenuation How full a surface is, against the pressure of the gas above it. The fraction of adsorption sites occupied on a surface that binds a molecule with 0.35 eV, against the gas pressure on a logarithmic axis, at 250, 300, 350 K. Each curve is θ = p/(p + p½) — Langmuir's isotherm — and each is the same S-shape moved along the axis: on a logarithmic pressure scale the coverage is a function of ln p alone, shifted by the binding energy over kT. The half-full pressure is 5.58 Pa at 250 K, 132 Pa at 300 K, 1341 Pa at 350 K. Warming the surface does not change the shape; it slides the curve to higher pressure by a factor that is a Boltzmann factor, e^(ε/kT), which is how a binding energy is read off a family of isotherms.

The site that fills like an electron level

A patch of catalyst holding a gas molecule, a haem in a muscle cell holding oxygen, an energy level in a metal holding an electron: each is a place that can hold one particle or none, in contact with a reservoir it trades particles with. Each fills according to the same function of one variable — how far the reservoir's chemical potential lies above the energy of the site. Langmuir's isotherm and the Fermi function are not analogous. They are the same law, and haemoglobin works because its sites break it.

thermodynamics · Chemical potential The centre of charge, and whether it moves. A pulsar of 1.46 and a white dwarf of 0.181 solar masses in a circular orbit, drawn twice about their fixed centre of mass (the cross). On the left each body carries a charge in exact proportion to its mass, as gravitational charge is under the equivalence principle: the centre of charge sits on the centre of mass and never moves, so the dipole moment is constant and nothing is radiated at dipole order. On the right the pulsar carries 60 per cent less charge per unit mass than its companion — an exaggerated mismatch, chosen so that the motion is visible at this scale — as a compact body's scalar charge would in a theory where its own binding energy couples differently: the centre of charge now circles the centre of mass at 12.6 per cent of the separation, once per orbit, and a dipole turning at the orbital frequency radiates.

The radiation a difference in falling would make

Gravitational waves start at the quadrupole because the equivalence principle removes the dipole: when every body's gravitational charge is its mass, the centre of charge is the centre of mass and cannot move. If a neutron star's own gravity fell even slightly differently, a binary containing one would radiate a dipole — louder than Einstein's quadrupole in slow orbits — and a pulsar with a white dwarf in an 8.5-hour orbit has been listening for it.

astrophysics · Equivalence principle A bound state that lives outside its well. The radial probability density of the one bound state of a square well of radius R, against distance in units of R, for four depths just past the one that first binds a state. In units of ε = ħ²/2μR², a state bound by 0.00589 ε has 93 per cent of its probability outside the well and a mean radius of 7.0 R; a state bound by 0.0397 ε has 82 per cent of its probability outside the well and a mean radius of 3.0 R; a state bound by 0.245 ε has 62 per cent of its probability outside the well and a mean radius of 1.5 R; a state bound by 1.4 ε has 35 per cent of its probability outside the well and a mean radius of 0.9 R. The more weakly the state is bound, the further its exponential tail reaches, and the less of it is where the force is.

The molecule that lives outside its own well

The harmonic picture describes the bottom of a well, where a molecule vibrates near the length of its bond. The last bound level lives at the top, and when it is bound by almost nothing it is not near the bond at all. Two helium atoms whose potential well bottoms out at three ångströms form a molecule whose atoms sit on average about fifty ångströms apart, most of the time where the force between them is negligible. Such a state forgets the shape of the well that made it. Its size, its binding and the way it scatters are all set by one number, and nothing inside the well can be read from them.

mechanics · Harmonic approximation The innermost level dives towards the sea. The energy of an atom's innermost (1s) electron, in units of the electron's rest energy mc², against the nuclear charge Z: dashed, Dirac's formula for a point nucleus, √(1 − (Zα)²); solid, the same equation solved for a nucleus of realistic size. Above +mc² lie the free electrons; below −mc², the filled sea of negative-energy states that Dirac's theory requires. For a point nucleus the level stops at Z = 137, where the formula fails. A real nucleus keeps it: it crosses zero near Z = 148 and reaches −mc² at Z ≈ 170, binding the electron by twice its own rest energy. Beyond that the level has nowhere left to go but into the sea.

The nucleus that would make positrons from nothing

Dirac's equation gives the innermost electron of an atom an energy that falls as the nuclear charge rises, and at 137 protons the formula fails outright. Real nuclei have a size, and with that correction the level does not stop: it falls through zero and on towards minus the electron's rest energy, the edge of the sea of states that Dirac's theory says are already full. It gets there at about 170 protons. Past that point an empty innermost level could be filled from the sea, and the vacuum round the nucleus would emit positrons of its own accord. No nucleus that large exists — but two uranium nuclei, for an instant, come close enough to be one.

quantum · Atomic spectra The parabola a family of isobars lies on. The atomic masses of the nuclei with 91 nucleons, as energy above the lightest, against the number that are protons, from the semi-empirical mass formula: a parabola with zirconium-91 at the bottom, which is where the stable isobar is. Left of the floor the nuclei have too many neutrons and β-decay towards it, each step turning a neutron into a proton and releasing the mass difference — 8.2 MeV from Kr, 5.8 MeV from Rb, 3.4 MeV from Sr, 1.0 MeV from Y — the steps shrinking as the floor nears, because the slope of a parabola does. Right of it the nuclei have too many protons and decay the other way, by capturing an electron or, where the drop exceeds 1.022 MeV, by emitting a positron: 6.2 MeV from Tc, 3.8 MeV from Mo, 1.4 MeV from Nb. Every arrow points downhill in mass, and nothing else decides its direction.

The parabola that decides which way a nucleus decays

A weak decay turns a neutron into a proton or a proton into a neutron and leaves the number of nucleons alone, so it moves a nucleus sideways among its isobars. Whether it can happen, and in which direction, is settled by weighing: a decay goes wherever the atom ends up lighter, by any amount at all, except that emitting a positron needs two electron masses more. Laid out against proton number, the masses of a family of isobars fall on a parabola — or, for an even number of nucleons, on two — and the shape of that parabola is why potassium-40 decays both ways and germanium-76 can only take two steps at once.

relativity · Mass-energy A mass that goes missing as light. For six captures of a slow neutron — ¹H to ²H, ¹²C to ¹³C, ⁵⁶Fe to ⁵⁷Fe, ²⁸Si to ²⁹Si, ³⁵Cl to ³⁶Cl and ³²S to ³³S — the mass that disappears, the atom plus the neutron less the atom formed, in thousandths of an atomic mass unit, against the energy released as gamma rays, in MeV, both computed from the tabulated atomic masses. All six lie on the line Δm = E/c², whose slope is one atomic mass unit per 931.494 MeV: ¹H → ²H loses 2.388 mu and releases 2.225 MeV; ¹²C → ¹³C loses 5.310 mu and releases 4.946 MeV; ⁵⁶Fe → ⁵⁷Fe loses 8.208 mu and releases 7.646 MeV; ²⁸Si → ²⁹Si loses 9.097 mu and releases 8.474 MeV; ³⁵Cl → ³⁶Cl loses 9.211 mu and releases 8.580 MeV; ³²S → ³³S loses 9.277 mu and releases 8.642 MeV. Drawn this way the line is a definition — the energies were computed from the masses. The experiment of 2005 measured the two quantities for silicon and sulphur independently, the masses as frequencies and the energies as angles, and asked whether they still fell on it.

The test of mass and energy that needs a ruler

The most direct test of E = mc² weighed two silicon atoms and measured a gamma ray. A silicon-28 nucleus that captures a slow neutron becomes silicon-29 and loses three ten-thousandths of its mass as gamma rays; weigh both atoms in a trap where they circle a magnetic field, measure the gamma rays by the angle at which a silicon crystal diffracts them, and the two sides agree to four parts in ten million. What the comparison needed is the interesting part: the recoil of the atom, a correction four hundred times larger than the precision, computed from the momentum relativity gives to light; angles of a few hundred picoradians; and a chain of constants tying a frequency and a ruler to the same number.

relativity · Mass-energy

Named alongside it

The objects these essays reach for when they reach for this one.

Equivalence principleE = mc²AbsorptionAttenuationScreeningSemi-empirical mass formulaStrong equivalence principleX-raysZero-point energyActivation barrierAtomic mass unitAtomic structure

All concepts